Math Insight

Elementary discrete dynamical systems problems, part 2

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  1. For the discrete dynamical system \begin{align*} z_{ t+1 } - z_t &= -0.3z_t^2 +1.2z_t -0.333\\ z_0 &= -1, \end{align*} find the equilibria and determine their stability analytically. Then, on the plot below, cobweb near each equilibrium to graphically verify your conclusions about stability.

  2. Consider the dynamical system \begin{align*} w_{ n+1} - w_n &= 1.7 w_n\left(1-\frac{ w_n }{ K }\right) \quad \text{for $n=0,1,2,3, \ldots$} \end{align*} where $K$ is a positive parameter.

    Find all equilibria and determine their stability.

    Does the stability of any of the equilibria depend on the value of $K$?

  3. For the following discrete dynamical system \begin{align*} x_{ t+1 } &= f(x_t)\\ x_0 &= -5, \end{align*} where $f(x) = 0.02 x^{3} + 0.04 x^{2} + 0.94 x$, the equilibria are $E=-3$, $E=0$, and $E=1$. For each equilibrium, determine the stability.

  4. For the following discrete dynamical system \begin{align*} z_{ t+1 } &= g(z_t)\\ z_0 &= -4, \end{align*} where $g(z) = - 0.09 z^{3} + 0.09 z^{2} + 2.53 z + 1.35$, the equilibria are $E=-3$, $E=-1$, and $E=5$. For each equilibrium, determine the stability analytically. Then, on the below graph, sketch tangent lines to $g$ at each equilibrium and cobweb near each equilibrium to graphically verify your conclusions about stability.

  5. Consider the dynamical system \begin{align*} x_{ t+1} - x_t &= r x_t\left(1-\frac{ x_t }{ 8000 }\right) \quad \text{for $t=0,1,2,3, \ldots$} \end{align*} where $r$ is a nonzero parameter.

    The system has two equilibria. What are they?

    For each equilibrium, determine the range of $r$ for which the equilibrium is stable.

  6. For the discrete dynamical system \begin{align*} z_{ n+1 } - z_n &= - 0.07 z_{n}^{3} - 0.42 z_{n}^{2} - 0.56 z_{n}\\ z_0 &= 7, \end{align*} determine the equilibria and their stability.

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Math 201, Spring 2017